populate a dropdown (or prop_search) with a string list
You're mistakingly asssuming that an EnumProperty can only hold integers. This works just fine:

import bpy

bpy.types.Object.my_prop = bpy.props.EnumProperty(
    items=[
        ('IDENTIFIER', 'The UI string', 'Tooltip'),
        ('JEMOEDER', 'Je moeder', 'Op je hoofd'),
    ])


For more info, see the bpy.props.EnumProperty API documentation.